A short note on generic initial ideals

The definition of a generic initial ideal includes the assumption x(1) > x(2) > ... > x(n). A natural question is how generic initial ideals change when we permute the variables. In the article [1, & sect;2], it is shown that the generic initial ideals are permuted in the same way when the variables in the monomial order are permuted. We give a different proof of this theorem. Along the way, we study the Zariski open sets which play an essential role in the definition of a generic initial ideal and also prove a result on how the Zariski open set changes after a permutation of the variables.